World Map



World Map with Confirmed and Death Cases

Total Confirmed for each Country

Total Deaths for each Country

Bar Chart



Bar Charts with descending order

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Bar Chart Confirmed

Bar Chart Table Deaths

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Table Confirmed and Deaths - Cumulated and Daily Cases

Bar Chart / Inhabitants



Bar Charts with descending order

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Bar Chart Confirmed per 100,000 Inhabitants

Bar Chart Deaths per 100,000 Inhabitants

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Table Confirmed and Deaths - Cumulated and Daily Cases per 100,000 Inhabitants

Cumulated and Daily Trend



Cumulated and Daily Cases over Time

Row

World

Row

Selected Countries

China

Austria

France

Germany

Italy

India

South Korea

Spain

United States of America

Germany - Confirmed and Deaths

Virus Spread on log10 scale



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Exponential Growth Evaluation

China and South Korea have slowed down significantly the exponential growth. Therefore, their lines in the chart with the log10 scale have no longer a significant slope.

Most other countries are still in a phase of more or less unchecked exponential growth. For Italy, the reduced exponential growth is reflected in a reduced slope of the cumulated cases.

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Virus Spread with log10 scale (since mid of Jan)

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Virus Spread / Inhabitants with log10 scale (since mid of Jan)

Exponential Growth



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Estimation spread of the Coronavirus with Linear Regression of log data

Exponential Growth and Doubling Time \(T\)

Exponential growth over time can be fitted by linear regression if the logarithms of the case numbers is taken. Generally, exponential growth corresponds to linearly growth over time for the log (to any base) data.

The semi-logorithmic plot with base-10 log scale for the Y axis shows functions following an exponential law \(y(t) = y_0 * a^{t/\tau}\) as straight lines. The time constant \(\tau\) describes the time required for y to increase by one factor of \(a\).

If e.g. the confirmed or death cases are growing in \(t-days\) by a factor of \(10\) the doubling time \(T \widehat{=} \tau\) can be calculated with \(a \widehat{=} 2\) by

\(T[days] = \frac {t[days] * log_{10}(2)} {log_{10}(y(t))-log_{10}(y_0)}\)

with

\(log_{10}(y(t))-log_{10}(y_0) = = log_{10}(y(t))/y_0) = log_{10}(10*y_0/y_0) = 1\)

and doubling time

\(T[days] = t[days] * log_{10}(2) \approx t[days] * 0.30\).

For Spain, Italy, Germany we have had a doubling time up to \(T \approx 9-12 days * 0.3 \approx 2.7 - 4 days !!\).

The doubling time \(T\) and the Forecast is calculated for following selected countries: Austria, France, Germany, Italy, India, South Korea, Spain, United States of America

Germany - Trend with Forecast on a linear scale

Forecast Plot - next 14 days

The plot shows the extreme forecast increase in case of unchecked exponentiell growth. The dark shaded regions show 80% rsp. 95% prediction intervals. These prediction intervals are displaying the uncertainty in forecasts based on the linear regression over the past 7 days.

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Comparison Exponential Growth

Germany - Example plot with ~linear slope on a log10 scale

Compare Exp vs Linear Growth



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Comparison Exponential vs. Linear Growth

The charts compare the different forecasts for an exponential rsp. linear growth model.

The dark shaded regions are indicating the \(80\%\) rsp. \(95\%\) prediction intervals. These prediction intervals are displaying the “pure” statistical uncertainty in forecasts based on the regression modles.

For doubling periods in the order of infectivity (RKI assumption: \(\sim9-10\) days, with great uncertainty), https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Modellierung_Deutschland.pdf?__blob=publicationFile), we no longer have exponential growth. Since the “old” infected cases are no longer infectious after these periods and we then have a constant infection rate with basic reproduction number \(R_0 \sim 1\).

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Cumulated Cases - Comparison Exponential and Linear Growth

Daily Cases - Comparison Exponential and Linear Growth

Doubling Time / Forecast



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Doubling Time and Forecast

The forecasted cases for the next 14 days are calculated ‘only’ from the linear regression of the logarithmic data and are not considering any effects of measures in place. In addition data inaccuracies are not taken into account, especially relevant for the confirmed cases.

Therefore the 14 days forecast is only an indication for the direction of an unchecked exponentiell growth.

Forecast (FC) with linear regression: Doubling Time (days), Forecasted cases tomorrow and Forecasted cases in 14 days
Country Case_Type T_doubling last_day FC_next_day FC_14days
Austria Confirmed 29.7 13’244 13’547 18’344
France Confirmed 8.0 118’781 137’186 420’565
Germany Confirmed 16.4 118’181 122’990 212’800
India Confirmed 4.2 6’725 8’342 70’127
Italy Confirmed 23.7 143’626 148’202 216’664
South Korea Confirmed 121.8 10’423 10’504 11’310
Spain Confirmed 16.9 153’222 160’611 274’036
United States of America Confirmed 8.2 461’437 508’471 1’524’848
World Confirmed 11.4 1’595’350 1’708’644 3’772’851
Austria Deaths 7.4 295 325 1’102
France Deaths 6.8 12’228 13’589 51’212
Germany Deaths 5.8 2’607 2’935 13’968
India Deaths 3.7 226 269 3’150
Italy Deaths 19.2 18’279 19’023 30’440
South Korea Deaths 25.2 204 210 300
Spain Deaths 13.0 15’447 16’434 32’938
United States of America Deaths 4.9 16’478 19’215 119’018
World Deaths 8.6 95’455 103’601 293’856

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Check of Forecast Accuracy

The forecast accuracy is checked by using the forecast method for the nine days before the past three days (training data). Subsequent forecasting of the past three days enables comparison with the real data of these days (test data).

The comparison is also an early indicator if the exponential growth is declining. However, possible changes in underreporting (in particular the proportion confirmed / actually infected) requires careful interpretation.

For doubling periods in the order of infectivity (RKI assumption: \(\sim9-10\) days, with great uncertainty), https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Modellierung_Deutschland.pdf?__blob=publicationFile), we no longer have exponential growth. Since the “old” infected cases are no longer infectious after these periods and we then have a constant infection rate with basic reproduction number \(R_0 \sim 1\).

Instead, we have “only” linear growth of the cumulative Confimred Cases and the Daily Confirmed Cases remain more or less constant or even decrease.

However, the basic reproduction number \(R_0\) is a product of the average number of contacts of an infectious person per day, the probability of transmission upon contacts and the average number of days infected people are infectious. With the current uncertainty of the average duration of the infectivity duration, \(R_0\) can therefore be estimated from the doubling time only to a very limited extent. See also: https://cmmid.github.io/topics/covid19/current-patterns-transmission/global-time-varying-transmission

Germany - Forecast Accuracy for past three days

Forecast



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Forecasting with lagged Predictors

The number of confirmed cases can be used as a time delayed predictor of the number of deaths. This will allow comclusions on the time period confirmed to death. More inportant the country specific case fatality rate (CFR, proportion of deaths from confirmed cases) indicates the country specific testing.

Overall a rough conclusion on the country specific proportion of infected to confirmed cases is feasible if the infection fatality rate (IFR, confiremd cases plus all asymptomatic and undiagnosed infections) is assumed to be country independent and the IFR is known (bottom of existing estimates \(\sim0.56\%\), assumption by RKI see https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Modellierung_Deutschland.pdf?__blob=publicationFile ).

Therefore an estimation of the CFR of \(0.06\) \((6\%)\) indicates an underreporting or lack of diagnostic confirmation by a by a factor of \(\sim10\). A CFR of \(0.20\) \((20\%)\) indicates an underreporting by a by a factor of \(\sim30\). This corresponds to RKI assumption of a underreporting by a factor of \(11-20\) (https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Steckbrief.html).

In the model paper RKI assumes for the

  • Incubation period \(\sim5-6\) days - Day of infection day until symptoms are upcoming)
  • Hospitalisation \(+4\) days - Admission to the hospital (if needed) after Incubation Period)
  • Average period to death \(+11\) - if the patient dies, it takes an average of \(11\) days after admission to the hospital

Depending on the country-specific test frequency (late or early tests), the

*lag_days - time from receipt of the confirmed test result to death, Confirmed to Death, is about \(11-13\) days.

Note: these methods are also used for example for advertising campaigns. The campaign impact on sales will be some time beyond the end of the campaign, and sales in one month will depend on the advertising expenditure in each of the past few months (see https://otexts.com/fpp3/lagged-predictors.html).

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Daily Confirmed and Death Cases

Lag days and Case Fatality Rate (CFR)

Lag days and CFR (proportion of deaths from confirmed cases)
Country lag_Confirmed CFR
Germany 12 0.04
Italy 12 0.06
Spain 10 0.03

Forecast residuals indicate quality of fit with Arima model:

ARIMA(Deaths ~ 0 + pdq(d = 0) + lag(Confirmed, lag_days - 2) + lag(Confirmed, lag_days - 1) + lag(Confirmed, lag_days) + lag(Confirmed, lag_days + 1) + lag(Confirmed, lag_days + 2)) .

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Daily Deaths depending on lagged Daily Confirmed Cases

Exampla Germany - White Noise of Forecast Residuals

References



Data Source

Data Source

Data files are provided by Johns Hopkins University on GitHub
https://github.com/CSSEGISandData/COVID-19/tree/master/csse_covid_19_data/csse_covid_19_time_series

  • Data files:
    • time_series_covid19_confirmed_global.csv
    • time_series_covid19_deaths_global
    • time_series_covid19_recovered_global.csv

The data are visualized on their excellent Dashboard
Johns Hopkins University Dashboard
https://coronavirus.jhu.edu/map.html